Christmas is almost here, so imagine you want to buy a good popular math book for your aunt (or whoever you want). Which book would you buy or recommend?

It would be nice if you could answer in the following way:

Title: The Poincaré Conjecture: In Search of the Shape of the UniverseAuthor: Donal O'Shea Short description: The history of the Poincaré Conjecture.
(Perhaps something like "difficulty level": + (no prior knowledge of math, as the book mentioned above), ++ (some prior knowledge of math is helpful), +++ (Roger Penrose: Road to Reality (?))

I hope this is appropriate for MO, since I think is of interest to mathematicians (at least for those who want to buy a popular math book for some aunt :-) ).

61 Answers
61

Ian Stewart is a very good mathematics popularizer and I understand why many of his books are in this list. But I really don't understand why none of Martin Gardner's books are listed here (at the time I'm writing this).

My favorite Martin Gardner book (actually, an updated collection of Scientific American Mathematical Games columns) is The Colossal Book of Mathematics. From the book's cover: "Number Theory, Algebra, Geometry, Probability, Topology, Game Theory, Infinity, and other topics of recreational mathematics." I would say its difficulty level is "+".

Brief Description: Stewart surveys the history of symmetry in mathematics (from ancient times to present day). There is particular emphasis on the insolvability of the quintic, the invention of Galois theory, and its impact on modern mathematics.

Short Description: A number of chapters covering lots of subjects: counting numbers (including the coining of the word "googol"); $\pi$, $i$, and $e$; geometries, plane and "fancy" ("Lobachevsky's Eiffel Towers and Riemann's Holland Tunnels"); puzzles; paradoxes; chance and probability; topology ("rubber sheet geometry"); calculus. Very much in the spirit of Martin Gardner columns, but from before they existed (the book was originally published in 1940; Gardner began writing his columns in 1956). You can see a preview at Google books.

Short Description: The google.books description is well written. A book for nonbelievers. I had this as a kid, and i remember its questions and pictures having an effect on me. How many sides does a banana have and how can 4 colours colour this map.

Short Description: This book is primarily about complex numbers. I can't remember it that well, but most of the book doesn't require calculus (though a fair part definitely does). Most of it is about the algebra and geometry of complex numbers and is thus accessible to the nonmathematician. The last chapter has a very fun introduction to complex analysis.

Short description: A mixture of Arabian Nights and a classical puzzlebook. The stories follow a brilliant youth who solves mathematical problems. A highly enjoyable read for anyone who likes oriental tales, and just the right combination of adventures and mathematics for all aunts and nephews out there without a mathematics degree.

Malba Tahan is a pen name of the Brazilian mathematician and writer Júlio César de Mello e Souza. The book was first published as O homem que calculava in 1949.

Wonder why it is that nobody has mentioned Lion-Hunting & Other Mathematical Pursuits? That book is one of my all-time favorites.

The book was edited by Gerald L. Alexanderson, the same individual behind the random walks of G. Pólya. Not only does the book contain the original article that launched the theory of big game hunting as a branch of mathematical research of its own, but also several of the subsequent contributions motivated by that 1938 groundbreaking paper of H. Pétard, e.g.:

If there is an even number of lions in the Sahara Desert we add a tame lion. Thus we may assume that the group of the Sahara lions is of odd order. This renders the situation capable of solution according to the work of Feit and Thompson.

It contains alot of introductory information on the spectrum
of mathematics with historical note. When you need a quick
perspective on branches of mathematics that are less familiar
to you, it is a place to go. The book is divided into sections
on famous mathematicians, theorems, and branches of mathematics
all of which cannot be held by even the most well versed
mathematician.

Short Review: "The Author considers these problems to be the type that two mathematical friends would pose to each other and discuss over a cup of coffee in a lounge. I agree with that premise, they are not too hard and there is a proof that is relatively easy to discover and even easier to understand. These problems satisfy all of the requirements for a good problem..."

This is one of the most entertaining recreation-mathematical books I've ever read. It does require some mathematical knowledge to be fully appreciated though :)

Description: (by Carroll himself in the preface) The writer’s intention was to embody in each Knot (like medicine so dexterously, but ineffectually, concealed in the jam of our early childhood) one or more mathematical questions — in Arithmetic, Algebra, or Geometry, as the case might be — for the amusement, and possible edification, of the fair readers of that magazine.

Of course, this will work as a gift for a niece better than a gift for an aunt, but it is one of those few books (the only other ones I know are by Smullyan and they've been mentioned already) that have both an entertaining story line and some actual mathematical challenges for the reader in them and that require no special mathematical training to read and enjoy.

Description: Biography of David Hilbert. Fascinating. Don't have to be D.H. to enjoy it. [the following is added from a duplicate answer - feel free to clean up -- ed.] A beautiful biography of a famous mathematician, showing the passion for knowledge represented by Hilbert and others at Goettingen around 1900. The book has a good description of Hilbert as a person, describes his family and coworkers and (sadly) the decay at the end of Hibert's life as the Nazis took over. I ask some of my undergraduate students to read this so that they can see the excitement of being in our profession. (The book is out of print, I think, but available via half.com or ebay.)

Second recommendation Also, Time-Life, ca. 1965, published one of it's picture books on math, titled of all things
Mathematics (as I recall). Check it out if you can find it. Lots of cool pictures. Great
for mathematically inclined high school kids.

Note added in edit: The aforementioned book from the Time-Life series also had great
pictures of mathematicians. As I recall, there was one of Eilenberg lying on the couch
of his Grenwich Village apartment, coat and tie still on. It was captioned with a quote which
went, if I recall correctly, "Sometimes I like to think riding on the subway, but mostly I
like to think lying down." Absolutely formative in my personal approach to math!

Title: Perfect Rigor: A Genius and the Mathematical Breakthrough of the Century

Author: Masha Gessen

Short Description: Story of Grigory Perelman based on information from people who interacted with him. A lot of interesting stuff about life in general and mathematics education in particular in Soviet Union around 1980's. There is only one or two short chapters where the author tries to explain the mathematics but if you skip those you don't need any serious mathematical skills.

Short description: (++) A book describing how Mathematics are founded, and tries to give a reasoning for the brain-work needed to do math.

From Amazon: The Mathematician's Brain poses a provocative question about the world's most brilliant yet eccentric mathematical minds: were they brilliant because of their eccentricities or in spite of them? In this thought-provoking and entertaining book, David Ruelle, the well-known mathematical physicist who helped create chaos theory, gives us a rare insider's account of the celebrated mathematicians he has known-their quirks, oddities, personal tragedies, bad behavior, descents into madness, tragic ends, and the sublime, inexpressible beauty of their most breathtaking mathematical discoveries.

Description : I read this while in high school, it's kind of a murder mystery, and it revolves around a family who has a mathematician friend who was trying to solve the Goldbach conjecture and Fermat's last theorem before he died. It includes many fun mathematics and is presented in a very light to read manner.

For sure this is an interesting book, first time I red first section then I was 18 I loved it. Second time I red it when I was 25-27 I hated it. Maybe it is time to read it again :)
–
DmitriJan 19 '10 at 0:21

Short description While not limited to mathematics, this is a great book which presents some subtle mathematical ideas in an intuitive non-technical way. I especially like the presentation of Cantor's infinite cardinality theory, which can be followed by anyone. Unfortunately, there are some mistakes. I seem to recall his presentation implicitly assumes the continuum hypothesis, but maybe he did that for clarity. It doesn't really detract from the book.

Short Description: Mathematical puzzle columns, (originally for the French version of Scientific American) some of which introduce interesting concepts from higher mathematics. Ian Stewart has written a lot in this vein (as have others, e.g. Martin Gardner), but this book is my favorite and is one of the mathematically meatiest of them (in my opinion).

I'd recomend "The Pea and the Sun, a mathematical paradox" which roughly explains the proof of Bannach-Tarski Theorem, which is a consequence of the axiom of choice and roughly says that you can break a pea in a finite number of pieces, tehn reassemble the pieces and get the sun

Short Description: A history of geometry of sorts. It's very well-written and was one of my favorite math books growing up. But you should read the Table of Contents to actually get an idea of what the book is about.

Title: Introduction to Mathematical Thinking: The Formation of Concepts in Modern Mathematics

Author: Friedrich Waismann

Short Description: This book, first published in 1936, gives an introduction to the philosophy of mathematics and the foundations of analysis. It shows how to construct number systems, some very elementary set theory and how to make differential calculus precise. The focus is on concepts. It includes a discussion of things like continuuous, nowhere differentiable functions and discusses why R and R2 are isomorphic as sets but not topologically. It even gives an very short and intuitive proof that R and R2 are not homeomorphic.

The philosophical discussion is, naturally, a litlle bit dated. It discusses formalism, logicism and intuitivism. The philosophy still makes for exciting historical reading. It is clear that this discussion was very lively when this book was written.

The level should youalify as ++, the book is demanding but can certainly be read before going to college.

I know this is a little late for Christmas, but nevertheless, I have a few (some of which have already been mentioned) books I've read that I've quite enjoyed. For the sake of brevity, I'll let you search the titles on Amazon for reviews and better descriptions.

Title: Everything & More: A Compact History of Infinity
Author: David Foster Wallace

Short description: "A marvellously lucid guide to the beauty and mystery of numbers"

This book is excellent for someone who wants to delve in to some basic, but interesting mathematics. It avoids very "popular mathematics" such as chaos theory and focuses in detail on "mundane" topics, such as dimension and estimation.

Title: The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity

Author: Amir D. Aczel

Short Description: Another book about the development of ideas about infinity. The central character is Cantor, of course, but it also looks at people before, like Bolzano and Galileo, and afterwards, including Godel and Cohen for their work on CH.

I read this in high school, having just a little calculus background, and got a lot out of it. It does try to work in the themes of "contemplation of infinity leads to insanity" and "infinity as religious insight" a bit, which might be drawbacks. But when I read it for the first time, I remember laughing out loud at how amazing the ideas involving infinte cardinalites and AC were, so at least those are presented well.

Short description (from New York Times Book Review): The real originality of PRISONER'S DILEMMA lies in its colorful synthesis of logical material and historical and biographical narration [which] takes us in parallel lines through cold war history, strategic games of the nuclear age and the life of von Neumann . . . Lively, open and multifaceted.

Indeed, the book can be read as a whole or just by following one of those "parallel lines". Depending of the line you choose, the level of difficulty would be + or ++!